What is the general solution for 3/(cos^2(x))=(7+4)/(cot(x)) ?

The general solution for 3/(cos^2(x))=(7+4)/(cot(x)) is x=(0.57693…)/2+pin,x= pi/2-(0.57693…)/2+pin

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3/(cos^2(x))=(7+4)/(cot(x))

Solution

\frac{3}{cos ^{2} ( x )} = \frac{7 + 4}{cot ( x )}

Solution

x = \frac{0.57693 \dots}{2} + πn, x = \frac{π}{2} - \frac{0.57693 \dots}{2} + πn

Degrees

x = 16.52786 \dots^{\circ} + 18 0^{\circ} n, x = 73.47213 \dots^{\circ} + 18 0^{\circ} n

Solution steps

\frac{3}{cos ^{2} ( x )} = \frac{7 + 4}{cot ( x )}

Subtract

\frac{7 + 4}{cot ( x )}

from both sides

\frac{3}{cos ^{2} ( x )} - \frac{11}{cot ( x )} = 0

Simplify

\frac{3}{cos ^{2} ( x )} - \frac{11}{cot ( x )} : \frac{3 cot ( x ) - 11 cos ^{2} ( x )}{cos ^{2} ( x ) cot ( x )}

\frac{3}{cos ^{2} ( x )} - \frac{11}{cot ( x )}

Least Common Multiplier of

cos^{2} (x), cot (x) : cos^{2} (x) cot (x)

cos^{2} (x), cot (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos^{2} (x)

cot (x)

= cos^{2} (x) cot (x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

cos^{2} (x) cot (x)

For

\frac{3}{cos ^{2} ( x )} :

multiply the denominator and numerator by

cot (x)

\frac{3}{cos ^{2} ( x )} = \frac{3 cot ( x )}{cos ^{2} ( x ) cot ( x )}

For

\frac{11}{cot ( x )} :

multiply the denominator and numerator by

cos^{2} (x)

\frac{11}{cot ( x )} = \frac{11 cos ^{2} ( x )}{cot ( x ) cos ^{2} ( x )}

= \frac{3 cot ( x )}{cos ^{2} ( x ) cot ( x )} - \frac{11 cos ^{2} ( x )}{cot ( x ) cos ^{2} ( x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 cot ( x ) - 11 cos ^{2} ( x )}{cos ^{2} ( x ) cot ( x )}

\frac{3 cot ( x ) - 11 cos ^{2} ( x )}{cos ^{2} ( x ) cot ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 cot (x) - 11 cos^{2} (x) = 0

Express with sin, cos

- 11 cos^{2} (x) + 3 cot (x)

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

= - 11 cos^{2} (x) + 3 \cdot \frac{cos ( x )}{sin ( x )}

Simplify

- 11 cos^{2} (x) + 3 \cdot \frac{cos ( x )}{sin ( x )} : \frac{- 11 cos ^{2} ( x ) sin ( x ) + 3 cos ( x )}{sin ( x )}

- 11 cos^{2} (x) + 3 \cdot \frac{cos ( x )}{sin ( x )}

Multiply

3 \cdot \frac{cos ( x )}{sin ( x )} : \frac{3 cos ( x )}{sin ( x )}

3 \cdot \frac{cos ( x )}{sin ( x )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{cos ( x ) \cdot 3}{sin ( x )}

= - 11 cos^{2} (x) + \frac{3 cos ( x )}{sin ( x )}

Convert element to fraction:

11 cos^{2} (x) = \frac{11 cos ^{2} ( x ) sin ( x )}{sin ( x )}

= - \frac{11 cos ^{2} ( x ) sin ( x )}{sin ( x )} + \frac{cos ( x ) \cdot 3}{sin ( x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 11 cos ^{2} ( x ) sin ( x ) + cos ( x ) \cdot 3}{sin ( x )}

= \frac{- 11 cos ^{2} ( x ) sin ( x ) + 3 cos ( x )}{sin ( x )}

\frac{3 cos ( x ) - 11 cos ^{2} ( x ) sin ( x )}{sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 cos (x) - 11 cos^{2} (x) sin (x) = 0

Factor

3 cos (x) - 11 cos^{2} (x) sin (x) : cos (x) (3 - 11 sin (x) cos (x))

3 cos (x) - 11 cos^{2} (x) sin (x)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

sin (x) cos^{2} (x) = cos (x) cos (x)

= 3 cos (x) - 11 cos (x) cos (x)

Factor out common term

cos (x)

= cos (x) (3 - 11 sin (x) cos (x))

cos (x) (3 - 11 sin (x) cos (x)) = 0

Solving each part separately

cos (x) = 0 or 3 - 11 sin (x) cos (x) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

General solutions for

cos (x) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

3 - 11 sin (x) cos (x) = 0 : x = \frac{arcsin ( \frac{6}{11} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

3 - 11 sin (x) cos (x) = 0

Rewrite using trig identities

3 - 11 sin (x) cos (x)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

= 3 - 11 \cdot \frac{sin ( 2 x )}{2}

3 - 11 \cdot \frac{sin ( 2 x )}{2} = 0

Move

3

to the right side

3 - 11 \cdot \frac{sin ( 2 x )}{2} = 0

Subtract

3

from both sides

3 - 11 \cdot \frac{sin ( 2 x )}{2} - 3 = 0 - 3

Simplify

- 11 \cdot \frac{sin ( 2 x )}{2} = - 3

- 11 \cdot \frac{sin ( 2 x )}{2} = - 3

Refine

- 11 \cdot \frac{sin ( 2 x )}{2} : - \frac{11 sin ( 2 x )}{2}

- 11 \cdot \frac{sin ( 2 x )}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{sin ( 2 x ) \cdot 11}{2}

- \frac{11 sin ( 2 x )}{2} = - 3

Multiply both sides by

2

- \frac{11 sin ( 2 x )}{2} = - 3

Multiply both sides by

2

- \frac{11 sin ( 2 x )}{2} \cdot 2 = - 3 \cdot 2

Simplify

- \frac{11 sin ( 2 x )}{2} \cdot 2 = - 3 \cdot 2

Simplify

- \frac{11 sin ( 2 x )}{2} \cdot 2 : - 11 sin (2 x)

- \frac{11 sin ( 2 x )}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{11 sin ( 2 x ) \cdot 2}{2}

Cancel the common factor:

2

= - 11 sin (2 x)

Simplify

- 3 \cdot 2 : - 6

- 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= - 6

- 11 sin (2 x) = - 6

- 11 sin (2 x) = - 6

- 11 sin (2 x) = - 6

Divide both sides by

- 11

- 11 sin (2 x) = - 6

Divide both sides by

- 11

\frac{- 11 sin ( 2 x )}{- 11} = \frac{- 6}{- 11}

Simplify

sin (2 x) = \frac{6}{11}

sin (2 x) = \frac{6}{11}

Apply trig inverse properties

sin (2 x) = \frac{6}{11}

General solutions for

sin (2 x) = \frac{6}{11}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

2 x = arcsin (\frac{6}{11}) + 2 πn, 2 x = π - arcsin (\frac{6}{11}) + 2 πn

2 x = arcsin (\frac{6}{11}) + 2 πn, 2 x = π - arcsin (\frac{6}{11}) + 2 πn

Solve

2 x = arcsin (\frac{6}{11}) + 2 πn : x = \frac{arcsin ( \frac{6}{11} )}{2} + πn

2 x = arcsin (\frac{6}{11}) + 2 πn

Divide both sides by

2

2 x = arcsin (\frac{6}{11}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arcsin ( \frac{6}{11} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arcsin ( \frac{6}{11} )}{2} + πn

x = \frac{arcsin ( \frac{6}{11} )}{2} + πn

Solve

2 x = π - arcsin (\frac{6}{11}) + 2 πn : x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

2 x = π - arcsin (\frac{6}{11}) + 2 πn

Divide both sides by

2

2 x = π - arcsin (\frac{6}{11}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

x = \frac{arcsin ( \frac{6}{11} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{arcsin ( \frac{6}{11} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn

x = \frac{arcsin ( \frac{6}{11} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

Show solutions in decimal form

x = \frac{0.57693 \dots}{2} + πn, x = \frac{π}{2} - \frac{0.57693 \dots}{2} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3/(cos^2(x))=(7+4)/(cot(x)) ?
The general solution for 3/(cos^2(x))=(7+4)/(cot(x)) is x=(0.57693…)/2+pin,x= pi/2-(0.57693…)/2+pin